Efficient implementation of interval matrix multiplication

International audienceThe straightforward implementation of interval matrix product suf- fers from poor efficiency, far from the performances of highly optimized floating-point implementations. In this paper, we show how to reduce the interval matrix multiplication to 9 floating-point matrix products - for performance issues - without sacrificing the quality of the result. We show that, compared to the straightforward implementation, the overes- timation factor is at most 1.18

Certification of a Numerical Result: Use of Interval Arithmetic and Multiple Precision

International audienceUsing floating-point arithmetic to solve a numerical problem yields a computed result, which is an approximation of the exact solution because of roundoff errors. In this paper, we present an approach to certify the computed solution. Here, "certify" means computing a guaranteed enclosure of the error between the computed, approximate, result and the exact, unknown result. We discuss an iterative refinement method: classically, such methods aim at computing an approximation of the error and they add it to the previous result to improve its accuracy. We add two ingredients: interval arithmetic is used to get an enclosure of the error instead of an approximation, and multiple precision is used to reach higher accuracy. We exemplify this approach on the certification of the solution of a linear system

Solving and Certifying the Solution of a Linear System

The Reliable Computing journal has no more paper publication, only free, electronic publication.International audienceUsing floating-point arithmetic to solve a linear system yields a computed result, which is an approximation of the exact solution because of roundoff errors. In this paper, we present an approach to certify the computed solution. Here, "certify" means computing a guaranteed enclosure of the error. Our method is an iterative refinement method and thus it also improves the computed result. The method we present is inspired from the verifylss function of the IntLab library, with a first step, using floating-point arithmetic, to solve the linear system, followed by interval computations to get and refine an enclosure of the error. The specificity of our method is to relax the requirement of tightness of the error, in order to gain in performance. Indeed, only the order of magnitude of the error is needed. Experiments show a gain in accuracy and in performance, for various condition number of the matrix of the linear system

Refining and verifying the solution of a linear system

International audienceThe problem considered here is to refine an approximate, numerical, solution of a linear system and simultaneously give an enclosure of the error between this approximate solution and the exact one: this is the verification step. Desirable properties for an algorithm solving this problem are accuracy of the results, complexity and performance of the actual implementation. A new algorithm is given, which has been designed with these desirable properties in mind. It is based on iterative refinement for accuracy, with well-chosen computing precisions, and uses interval arithmetic for verification

Further Discussion On Applying Case Study Method In Investment Law Course – A Case Of Education Investment In Vietnam

Purpose: The aim of this study is to focus on clarifying using case study method to improve taching law course. Currently, many universities around the world are applying a new teaching method. It can help students absorb knowledge more efficiently and faster. That is the case study. Using a Case Study allows a deeper understanding of cases as a unit, through comparing the similarities and differences of individual cases. Design/methodology/approach: The study uses qualitative analysis, synthesis and inductive methods, together with studies and references from CPTPP, FTA Agreement, as well as Regulations on assurance of business investment activities. Findings: The application of this method to teaching helps learners to approach very applicable situations. The situations or problems close to the theory will also help them to systematize all the knowledge they have learned to apply to solve the assigned problem. Not only use the knowledge learned, but also learn related knowledge outside the program to solve the problem. Case Study is an extremely interesting and interesting learning method, helping us to associate reality and apply it to solve problems and help us remember longer. If you really understand what a case study is, then apply it to your life and work. The results show the regulations on guaranteeing ownership rights as well as ensuring investors' business investment activities have been clearly stipulated in the 2020 Labor Law of Vietnam. These regulations have concretized the constitutional principle of investors' freedom to invest and do business in industries that are not prohibited by la

FPGA-Specific Arithmetic Optimizations of Short-Latency Adders

International audienceInteger addition is a pervasive operation in FPGA designs. The need for fast wide adders grows with the demand for large precisions as, for example, required for the implementation of IEEE-754 quadruple precision and eliptic-curve cryptography. The FPGA realization of fast and compact binary adders relies on hardware carry chains. These provide a natural implementation environment for the ripple-carry addition (RCA) scheme. As its latency grows linearly with the operand width, wide additions call for acceleration, which is quite reasonably achieved by addition schemes built from parallel RCA blocks. This study presents FPGA-specific arithmetic optimizations for the mapping of carry-select/increment adders targeting the hardware carry chains of modern FPGAs. Different trade-offs between latency and area are presented. The proposed architectures represent attractive alternatives to deeply pipelined RCA schemes

Tradeoffs between Accuracy and Efficiency for Optimized and Parallel Interval Matrix Multiplication

International audienceInterval arithmetic is mathematically defined as set arithmetic. For implementation issues, it is necessary to detail the representation of intervals and to detail formulas for the arithmetic operations. Two main representations of intervals are considered here: inf-sup and midrad. Formulas for the arithmetic operations, using these representations, are studied along with formulas that trade off accuracy for efficiency. This tradeo ff is particularly blatant on the example of interval matrix multiplication, implemented using floating-point arithmetic: depending on the chosen formulas, the effi ciency as well as the accuracy can vary greatly in practice, and not necessarily as predicted by the theory. Indeed, theoretical predictions are often based on exact operations, as opposed to floating-point operations, and on operations count, as opposed to measured execution time. These observations and the recommendations that ensue are further obfuscated by considerations on memory usage, multithreaded computations. . . when these algorithms are implemented on parallel architectures such as multicores.L'arithmétique par intervalles et une arithmétique sur les ensembles. Pour pouvoir l'implanter, il faut détailler d'une part la représentation choisie pour les intervalles et d'autre part les formules, dépendant de cette représentation, pour les opérations arithmétiques. Essentiellement deux représentations des intervalles sont considérées ici : la représentation inf-sup (par les extrémités) et la représentation mid-rad (par le centre et le rayon). Différentes formules pour les opérations arithmétiques sont présentées, qui offrent différents compromis entre la précision du résultat et la quantité de calculs à effectuer. Ces compromis sont encore plus flagrants quand on considère l'utilisation de ces opérations pour effectuer des produits de matrices à coefficients intervalles, implantés en utilisant l'arithmétique flottante : selon les formules choisies, les performances ainsi que la précision peuvent différer grandement en pratique, mais pas nécessairement comme le prédit la théorie. En effet, les prédictions théoriques sont basées sur l'hypothèse d'une arithmétique sous-jacente qui est exacte et non flottante, ainsi que sur un décompte d'opérations, ce qui ne correspond pas directement aux mesures des temps d'exécution. Ces observations et les recommandations qui en découlent dépendent en outre de considérations sur l'utilisation mémoire, la présence de calculs multithreadés etc. lorsque l'on considère des implantations parallèles sur des architectures telles que des multi-cœurs

Basis for the development of criteria for monitoring the implementation of land laws in Vietnam

This paper is one of the newest studies which focuses on the issue of monitoring the land law implementation in Vietnam. The research clarifies the basic concepts related to monitoring the implementation of land law comprehensively. Also, in the context of Vietnamese legislation system has not regulated a sufficient criteria system in this issue, the author developed a suitable criteria system in order to enhance the efficiency of monitoring the implementation of land law in Vietnam. To reach the goal, the author used both of qualititative and quantitative methods to carrying out the research. Especially, the author collected the huge amount of data which covered doing structured interviews from 23 provinces in Vietnam to gain the results comprehensively and objectively. The author also worked in consultation with people working in land sector and relevant experts by doing unstructured interviewed. The experts had an opportuinity to express their views on the status quo on land and suggest the content of the criterias in monitoring the implementation of land law. The results reveal that the majority of interviewees point out difficulties related to the lack of the criteria and suggest that they should be developed immediately. The system of criterias in the research is expected to become the basis for monitoring the implementation of land law in Vietnam effectively

The Role of Tax in Managing Offshore Fishing Activities: An Application of Bioeconomic Model

The paper’s purpose is to light up the role of tax in managing offshore fishing activities. It was conducted by considering two management regimes; (1) assumes that the number of fishing firms is finite and defined as a restricted access regime, and (2) the number of fishing firms is infinite, which is regarded as an open access regime, to enter into a coastal nation’s fisheries. Each management regime is linked to two policy goals; namely, the maximum sustainable biomass yield and maximum sustainable economic yield goal. The Verhulst Schaefer model and Wachsman model were employed to estimate the biological parameters and economic analysis. The results of empirical analysis indicated that a coastal nation could control the effort and stock through adjusting the fishing fee that can be determined as a tax policy. More especially, if a coastal nation didn’t charge any fee level on fishing firms, even though the number of firms was restricted, the stock still was under acceptable level, which leads to the depletion and potential collapse risk of fisheries. Additionally, a coastal nation could adopt one of four fee levels such as rMSBY (0.3378), rMSEY (0.5071), r’MSBY (0.3623), and r’MSEY (0.5165) depending on each management regime and policy goal selected. However, it would be better if the coastal nation selects the open access regime, and then charges on fishing firms at level of r’MSBY or r’MSEY. Keywords: Tax, Fishing Management regime, Policy goal Classification: JEL, Q 2

The Optimal Point for Fiscal Decentralization

The paper’s purpose is to shed more light on the impact of fiscal decentralization on the economy and determine whether or not a tipping point can be identified as an optimal point of fiscal decentralization. To do so, we proposed a new theoretical model to link two measurements of fiscal decentralization such as fiscal autonomy and fiscal importance to provincial GDP, and then apply our model with panel data to provincial GDP of Vietnam over ten years and across 56 provinces to test the significance of the impact of fiscal decentralization on the economy and compute the optimal point of fiscal decentralization. Generalized linear model with maximum likelihood method was applied to estimate coefficients in the analytical model. The results of empirical analysis indicated that our model is statistically significant and there exists an optimal point for fiscal decentralization with value captured is 7.33 of fiscal autonomy index and 0.25 of fiscal importance index. Additionally, the study also investigated that the fiscal decentralization would become a positively influential element on the economy, if the degree of fiscal decentralization underlies the optimal point. If the degree of fiscal decentralization exceeds the optimal point, however, it would affect  negatively on the economy. Keyword: Fiscal decentralization, Optimal point, Provincial GD